Intro

One of my big projects this past year was preparing to my first Kaggle competition - both the NCAA Men's and Women's basketball tournaments! Unfortunately, both tournaments were cancelled due to the Covid-19 pandemic. Regardless, working through my different ideas was a great way to learn some of the basics and nuances of training an ML model effectively.

I tried a few different types of models including a simple 2 layer neural net using FastAI and an ensemble model (XGBoost). These were relatively comprable and both required a similar level of feature engineering to get a good result (I think due to the limited data set - ~64 games per year since 2003 with the detailed game data). The results were fine, but I was wondering if there was a solution that could allow for less feature engineering and still give a reasonable result.

Enter embeddings... in this notebook I'm going to explore a simple Keras implementation of embeddings to represent each NCAA team (e.g. Duke 2019 $\neq$ Duke 2020). For now, I will use those embeddings to perform some exploratory analysis to verify that they have learned useful features and in a following notebook I will use these embeddings as input features to an NCAA tournament model. This work was inspired by this Kaggle notebook. Though the predictions aren't spectacular, I think the exploratory analysis shows that embeddings could be a useful feature if paired with more detailed data and a model refined to predict NCAA tournament games as opposed to regular season games.

What you will see in this notebook at a high level:

  • Brief data prep - we are only using wins/losses, points, home/away, and team IDs as inputs to the model. *This work will later be expanded to incorporate more advanced statistics.
  • Model build - This model is being built with the sole purpose of generating useful embeddings. To achieve that we are training the model to be predictive of features that we would ordinarily use as feature inputs to a real tournament model (in this case, regular season wins and losses).
  • Training and validation - the model is trained using only regular season data from all years and is validated on a secondary set of tournament data (NIT). This is difficult because we have a slight mismatch between our training and validation data. The validation data is generally similar and likely more representative of the real NCAA tournament.
  • Sense check and exploratory analysis - First thing is to check that predictions from the model are sensisble, but what we really care about is the embeddings. Do they carry more useful information than simple aggregations of the data they represent? In short, Yes!

Packages and Data

I'll be implementing this in Keras. My previous attempt using FastAI was quick and easy. Using embeddings for categorical data made the FastAI model a bit more elegant than XGBoost. However, we need two input variables (team 1 and team 2) to call the same embeddings matrix in this solution. FastAI can't do that out of the box and so I get to venture into the world of building my own in Keras. I plan on going one step deeper and building my final tournament model with TensorFlow.

#collapse_hide
from pathlib import Path
import numpy as np
import pandas as pd
np.random.seed(13)
import tensorflow as tf
import keras as k
from keras.models import Model
from keras.layers import Dense, Input, Dropout, Activation, Multiply, Lambda, Concatenate, Subtract, Flatten
from keras.layers.embeddings import Embedding
from keras.initializers import glorot_uniform, glorot_normal
from keras.optimizers import Adam
import matplotlib.pyplot as plt
from scipy import stats
from sklearn.manifold.t_sne import TSNE
import altair as alt
np.random.seed(13)
Using TensorFlow backend.

I will be training the embeddings using total point and point differential from each game. Because the training doesn't require the more detailed NCAA data set, we can train using NCAA data all the way back to 1985. Hopfully this will make the weights of the other layers more robust. Depending on the embedding results, the final tournament model could also be trained back to 1985. Let's preview the first few rows of that regular season data here:

#collapse_hide
dataLoc=Path('./data/2020-05-04-NCAA-Embeddings/google-cloud-ncaa-march-madness-2020-division-1-mens-tournament/MDataFiles_Stage2/')

df_teams = pd.read_csv(dataLoc/'MTeams.csv')
teams_dict = df_teams[['TeamID','TeamName']].set_index('TeamID').to_dict()['TeamName']

df_regSeason_data = pd.read_csv(dataLoc/'MRegularSeasonCompactResults.csv')
df_regSeason_data.head() # cols = Season,DayNum,WTeamID,WScore,LTeamID,LScore,WLoc,NumOT
Season DayNum WTeamID WScore LTeamID LScore WLoc NumOT
0 1985 20 1228 81 1328 64 N 0
1 1985 25 1106 77 1354 70 H 0
2 1985 25 1112 63 1223 56 H 0
3 1985 25 1165 70 1432 54 H 0
4 1985 25 1192 86 1447 74 H 0

I want to be able to validate that the embedding training is going in the right direction. For the embedding training I will use the secondary tournament data. This allows us to avoid using the NCAA tournament data that we need for training/testing later, but still get a sense that the embeddings are useful. Here is a preview of that data:

#collapse_hide
df_otherTourney_data = pd.read_csv(dataLoc/'MSecondaryTourneyCompactResults.csv').drop(columns='SecondaryTourney')
df_otherTourney_data.head() # cols = Season,DayNum,WTeamID,WScore,LTeamID,LScore,WLoc,NumOT
Season DayNum WTeamID WScore LTeamID LScore WLoc NumOT
0 1985 136 1151 67 1155 65 H 0
1 1985 136 1153 77 1245 61 H 0
2 1985 136 1201 79 1365 76 H 0
3 1985 136 1231 79 1139 57 H 0
4 1985 136 1249 78 1222 71 H 0

Embeddings will be defined by the columns 'Season', 'WTeamID', and 'LTeamID'. 'WScore' and 'LScore' will be augmented slightly to be the predicted values and the game location will also be included as an embedding.

#collapse_hide
# Create team encoding that differentiates teams by year and school
def newTeamID(df):
    # df = df.sample(frac=1).reset_index(drop=True)
    df['Wnewid'] = df['Season'].astype(str) + df['WTeamID'].astype(str)
    df['Lnewid'] = df['Season'].astype(str) + df['LTeamID'].astype(str)
    return df

df_regSeason_data = newTeamID(df_regSeason_data)
df_otherTourney_data = newTeamID(df_otherTourney_data)

def idDicts(df):
    newid_W = list(df['Wnewid'].unique())
    newid_L = list(df['Lnewid'].unique())
    ids = list(set().union(newid_W,newid_L))
    ids.sort()
    oh_to_id = {}
    id_to_oh = {}
    for i in range(len(ids)):
        id_to_oh[ids[i]] = i 
        oh_to_id[i] = ids[i]

    return oh_to_id, id_to_oh

oh_to_id, id_to_oh = idDicts(df_regSeason_data)    

# add training data in swapped format so network sees both wins and losses
def swapConcat_data(df):

    df['Wnewid'] = df['Wnewid'].apply(lambda x: id_to_oh[x])
    df['Lnewid'] = df['Lnewid'].apply(lambda x: id_to_oh[x])

    loc_dict = {'A':-1,'N':0,'H':1}
    df['WLoc'] = df['WLoc'].apply(lambda x: loc_dict[x])

    swap_cols = ['Season', 'DayNum', 'LTeamID', 'LScore', 'WTeamID', 'WScore', 'WLoc', 'NumOT', 'Lnewid', 'Wnewid']

    df_swap = df[swap_cols].copy()

    df_swap['WLoc'] = df_swap['WLoc']*-1

    df.columns = [x.replace('WLoc','T1_Court')
                   .replace('W','T1_')
                   .replace('L','T2_') for x in list(df.columns)]

    df_swap.columns = df.columns

    df = pd.concat([df,df_swap])

    df['Win'] = (df['T1_Score']>df['T2_Score']).astype(int)
    df['Close_Game']= abs(df['T1_Score']-df['T2_Score']) <3
    df['Score_diff'] = df['T1_Score'] - df['T2_Score']
    df['Score_diff'] = df['Score_diff'] - (df['Score_diff']/df['Score_diff'].abs())
    df['T2_Court'] = df['T1_Court']*-1
    df[['T1_Court','T2_Court']] = df[['T1_Court','T2_Court']] + 1

    cols = df.columns.to_list()

    df = df[cols].sort_index()
    df.reset_index(drop=True,inplace=True)


    return df

df_regSeason_full = swapConcat_data(df_regSeason_data.copy().sort_values(by='DayNum'))
df_otherTourney_full = swapConcat_data(df_otherTourney_data.copy())

# Convert to numpy arrays in correct format
def prep_inputs(df,id_to_oh, col_outputs):
    Xteams = np.stack([df['T1_newid'].values,df['T2_newid'].values]).T
    Xloc = np.stack([df['T1_Court'].values,df['T2_Court'].values]).T

    if len(col_outputs) <2:
        Y_outputs = df[col_outputs].values
        Y_outputs = Y_outputs.reshape(len(Y_outputs),1)
    else:
        Y_outputs = np.stack([df[x].values for x in col_outputs])

    return [Xteams, Xloc], Y_outputs

X_train, Y_train = prep_inputs(df_regSeason_full, id_to_oh, ['Win','Score_diff'])
X_test, Y_test = prep_inputs(df_otherTourney_full, id_to_oh, ['Win','Score_diff'])

# Normalize point outputs - Win/loss unchanged
def normalize_outputs(Y_outputs, stats_cache=None):
    if stats_cache == None:
        stats_cache = {}
        stats_cache['mean'] = np.mean(Y_outputs,axis=1)
        stats_cache['var'] = np.var(Y_outputs,axis=1)
    else: pass
    
    numOut = Y_outputs.shape[0]
    Y_normout = (Y_outputs-stats_cache['mean'].reshape((numOut,1)))/stats_cache['var'].reshape((numOut,1))

    return Y_normout, stats_cache

Y_norm_train, stats_cache_train = normalize_outputs(Y_train,None)
Y_norm_test, _ = normalize_outputs(Y_test,stats_cache_train)
Y_norm_train[0,:] = Y_train[0,:]
Y_norm_test[0,:] = Y_test[0,:]

Building the model

This model is built with two input types - home/away flags and team IDs. Each input is repeated for each team and is fed through a location embedding layer and a team embedding layer. A school's embeddings are separate season to season. It would nice to be able to cary some dependency from year to year, but that is completely disregarded here for simplicity. The location embedding is 1-dimensional and multiplied by each team's embedding vector element by element. The team embeddings are separately fed through the same two-layers before being subtracted. This subtracted layerinally connect to two output layers - one 'softmax' for win/loss prediction and one dense layer with no activation for point prediction.

#collapse_show
# build model

tf.keras.backend.clear_session()

def NCAA_Embeddings_Joint(nteams,teamEmb_size):
    team_input = Input(shape=[2,],dtype='int32', name='team_input')
    X_team = Embedding(input_dim=nteams, output_dim=teamEmb_size, input_length=2, embeddings_initializer=glorot_uniform(), name='team_encoding')(team_input)

    loc_input = Input(shape=[2,],dtype='int32', name='loc_input')
    X_loc = Embedding(input_dim=3, output_dim=1, input_length=2, embeddings_initializer=glorot_uniform(), name='loc_encoding')(loc_input)
    X_loc = Lambda(lambda z: k.backend.repeat_elements(z, rep=teamEmb_size, axis=-1))(X_loc)
    
    X = Multiply()([X_team,X_loc])
    X = Dropout(rate=.5)(X)
    T1 = Lambda(lambda z: z[:,0,:])(X)
    T2 = Lambda(lambda z: z[:,1,:])(X)

    D1 = Dense(units = 20, use_bias=True, activation='tanh')
    DO1 = Dropout(rate=.5)

    D2 = Dense(units = 10, use_bias=True, activation='tanh')
    DO2 = Dropout(rate=.5)

    X1 = D1(T1)
    X1 = DO1(X1)

    X1 = D2(X1)
    X1 = DO2(X1)

    X2 = D1(T2)
    X2 = DO1(X2)

    X2 = D2(X2)
    X2 = DO2(X2)

    X_sub = Subtract()([X1,X2])

    output_w= Dense(units = 1, use_bias=False, activation='sigmoid', name='win_output')(X_sub)
    output_p= Dense(units = 1, use_bias=False, activation=None, name='point_output')(X_sub)


    model = Model(inputs=[team_input, loc_input],outputs=[output_w,output_p],name='ncaa_embeddings_joint')

    return model

mymodel = NCAA_Embeddings_Joint(len(id_to_oh),8)
mymodel.summary()
WARNING:tensorflow:From /Users/ryanarmstrong/opt/miniconda3/envs/ds37/lib/python3.7/site-packages/tensorflow/python/ops/resource_variable_ops.py:435: colocate_with (from tensorflow.python.framework.ops) is deprecated and will be removed in a future version.
Instructions for updating:
Colocations handled automatically by placer.
Model: "ncaa_embeddings_joint"
__________________________________________________________________________________________________
Layer (type)                    Output Shape         Param #     Connected to                     
==================================================================================================
loc_input (InputLayer)          (None, 2)            0                                            
__________________________________________________________________________________________________
team_input (InputLayer)         (None, 2)            0                                            
__________________________________________________________________________________________________
loc_encoding (Embedding)        (None, 2, 1)         3           loc_input[0][0]                  
__________________________________________________________________________________________________
team_encoding (Embedding)       (None, 2, 8)         92752       team_input[0][0]                 
__________________________________________________________________________________________________
lambda_1 (Lambda)               (None, 2, 8)         0           loc_encoding[0][0]               
__________________________________________________________________________________________________
multiply_1 (Multiply)           (None, 2, 8)         0           team_encoding[0][0]              
                                                                 lambda_1[0][0]                   
__________________________________________________________________________________________________
dropout_1 (Dropout)             (None, 2, 8)         0           multiply_1[0][0]                 
__________________________________________________________________________________________________
lambda_2 (Lambda)               (None, 8)            0           dropout_1[0][0]                  
__________________________________________________________________________________________________
lambda_3 (Lambda)               (None, 8)            0           dropout_1[0][0]                  
__________________________________________________________________________________________________
dense_1 (Dense)                 (None, 20)           180         lambda_2[0][0]                   
                                                                 lambda_3[0][0]                   
__________________________________________________________________________________________________
dropout_2 (Dropout)             (None, 20)           0           dense_1[0][0]                    
                                                                 dense_1[1][0]                    
__________________________________________________________________________________________________
dense_2 (Dense)                 (None, 10)           210         dropout_2[0][0]                  
                                                                 dropout_2[1][0]                  
__________________________________________________________________________________________________
dropout_3 (Dropout)             (None, 10)           0           dense_2[0][0]                    
                                                                 dense_2[1][0]                    
__________________________________________________________________________________________________
subtract_1 (Subtract)           (None, 10)           0           dropout_3[0][0]                  
                                                                 dropout_3[1][0]                  
__________________________________________________________________________________________________
win_output (Dense)              (None, 1)            10          subtract_1[0][0]                 
__________________________________________________________________________________________________
point_output (Dense)            (None, 1)            10          subtract_1[0][0]                 
==================================================================================================
Total params: 93,165
Trainable params: 93,165
Non-trainable params: 0
__________________________________________________________________________________________________

Training the model

The model is trained using regular season data and validated using secondary tournament data (not the 'Big Dance'). The weights of the two losses are adjusted so that they propogate a similar amount of error backward. Because the point differential data has been normalized, the losses are multiple orders of magnitude less than the log loss metric for wins/losses.

#collapse_show
# Joint model
optimizer = Adam(learning_rate=.01, beta_1=0.9, beta_2=0.999, amsgrad=False)
mymodel.compile(loss=['binary_crossentropy','logcosh'],
                loss_weights=[.5,400],
                optimizer=optimizer,
                metrics = ['accuracy'])
numBatch = round(X_train[0].shape[0]/50)
results = mymodel.fit(X_train, [*Y_norm_train], validation_data=(X_test, [*Y_norm_test]), epochs = 30, batch_size = numBatch,shuffle=True, verbose=False)
WARNING:tensorflow:From /Users/ryanarmstrong/opt/miniconda3/envs/ds37/lib/python3.7/site-packages/tensorflow/python/ops/math_ops.py:3066: to_int32 (from tensorflow.python.ops.math_ops) is deprecated and will be removed in a future version.
Instructions for updating:
Use tf.cast instead.

#collapse_hide
accuracy = results.history['win_output_accuracy']
val_accuracy = results.history['val_win_output_accuracy']
loss = results.history['win_output_loss']
val_loss = results.history['val_win_output_loss']
# summarize history for accuracy
plt.plot(accuracy)
plt.plot(val_accuracy)
plt.title('model accuracy')
plt.ylabel('accuracy')
plt.xlabel('epoch')
plt.legend(['train', 'test'], loc='upper left')
plt.show()
# summarize history for loss
plt.plot(loss)
plt.plot(val_loss)
plt.title('model loss')
plt.ylabel('loss')
plt.xlabel('epoch')
plt.legend(['train', 'test'], loc='upper left')
plt.show()
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Results

The crossplots for point differential and win/loss are generally well behaved. The loss and accuracy of the model are not great in comparison to results from the Kaggle competition. For reference, anything below a loss of 0.5 would be considered fantastic and flipping a coin would give you a loss of about 0.69. We see a similar effect in the point spread prediction with a rather loose correlation of 0.47.

If the goal of this project was to have the best model for predicting the winner of an NCAA tournament game we would be failing (especially considering only the best play in the tournament - making predictions even harder). However, the goal here was to train embeddings not to get accurate predictions. Instead, we are using regular season data to train an embedding set that is representative of each team. We have only trained on wins/losses and points in this case, which might limit the utility of the features. Converserly, we will see in the next section that we have achieved a richer representation of the raw win/loss data than simply aggregating by teams.

#collapse_hide
def transform_y(preds,stats_cache):
    preds = stats_cache['var'][1] * preds + stats_cache['mean'][1]
    return preds

preds = mymodel.predict(X_test)

tmp=0

x = transform_y(preds[1],stats_cache_train).reshape(-1)
y = transform_y(Y_norm_test[1],stats_cache_train).reshape(-1)


print('Pearson coefficient: ', round(stats.pearsonr(x, y)[0]*100)/100)
plt.scatter(x, y, alpha=0.08)
# plt.title('Scatter plot pythonspot.com')
plt.xlabel('Predicted point difference')
plt.ylabel('Actual point difference')
plt.show()
Pearson coefficient:  0.47
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#collapse_hide
x = preds[0].reshape(-1)

plt.hist(x,bins=100)
# plt.title('Scatter plot pythonspot.com')
plt.xlabel('Predicted Win Probability')
plt.ylabel('Count')
plt.show()
<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">

One notable aspect of the point prediction result is that the predictions are perfectly symmetrical. The network is able to give consistent predictions for "Team A vs. Team B" and "Team B vs. Team A" because the neural network is set up to treat the input features consistently for each team. Other ML models, such as XGBoost, treat the feature inputs of Team 1 and Team 2 differently, which can result in varying predictions when the teams are swapped. This can be an issue even when training sets contain matchups and swapped matchups as documented in this discussion.

Exploratory Analysis

Let's take a look at some comparisons between our embeddings (mapped non-linearly into 2D space by T-SNE) vs. the aggregated point differential and win percentage of each team. All of these plots (excluding the final plot for 2020) will only include teams that were included in the NCAA tournament that year.

We'll color the scatter plots by a few different factors:

  • Highlighting tournament winners
  • Tournament seed number
  • Number of tournament games won
  • Biggest opening weekend upsets according to this article

Here is a preview of the data that will be fed into the visualizations:

#collapse_hide
embeddings = mymodel.layers[3].get_weights()[0]

t = TSNE(n_components=2)
embed_tsne = t.fit_transform(embeddings)

df_regSeason_full['T1_TeamName'] = df_regSeason_full['T1_TeamID'].apply(lambda x: teams_dict[x]) + '-' + df_regSeason_full['Season'].astype(str)
df_agg=df_regSeason_full.groupby('T1_TeamName').mean()
df_agg.reset_index(inplace=True,drop=False)

df_agg[['T1_TeamName','Win','Score_diff']]
df_agg.drop(columns='Season',inplace=True)

df_tourney_data = pd.read_csv(dataLoc/'MNCAATourneyCompactResults.csv')
df_tourney_data['WTeamName'] = df_tourney_data['WTeamID'].apply(lambda x: teams_dict[x]) + '-' + df_tourney_data['Season'].astype(str)
df_tourney_data['Wins'] = 0
df_wins = df_tourney_data[['WTeamName','Wins']].groupby('WTeamName').count()
tourneyWinners = [df_tourney_data.loc[df_tourney_data['Season']==s,'WTeamName'].values[-1] for s in df_tourney_data['Season'].unique()]

df_seeds = pd.read_csv(dataLoc/'MNCAATourneySeeds.csv')
df_seeds['TeamName'] = df_seeds['TeamID'].apply(lambda x: teams_dict[x]) + '-' + df_seeds['Season'].astype(str)
df_seeds['Seed'] = df_seeds['Seed'].str.extract(r'(\d+)')
df_seeds['WonTourney'] = df_seeds['TeamName'].apply(lambda x: True if x in tourneyWinners else False)
df_seeds = df_seeds[['TeamName','Seed','WonTourney']]

df_upsets = pd.read_csv('./data/2020-05-04-NCAA-Embeddings/Upsets.csv')
df_upsets['David']=df_upsets['David']+'-'+df_upsets['Season'].astype(str)
df_upsets['Goliath']=df_upsets['Goliath']+'-'+df_upsets['Season'].astype(str)
upsets = {}
for ii in df_upsets['David'].unique():
    upsets[ii] = 'Surprise'
for ii in df_upsets['Goliath'].unique():
    upsets[ii] = 'Bust'
df_seeds = pd.merge(left=df_seeds, right=df_wins, how='left', left_on='TeamName',right_index=True)
df_seeds['Wins'].fillna(0,inplace=True)

def upset(x):
    try:
        y = upsets[x]
    except:
        y = None
    return y
df_seeds['Upset'] = df_seeds['TeamName'].apply(lambda x: upset(x))

df = pd.DataFrame(embed_tsne,columns=['factor1','factor2'])
df['TeamName'] = [str(teams_dict[int(oh_to_id[x][-4:])]) + '-' + oh_to_id[x][:4] for x in df.index]
df['Season'] = [int(oh_to_id[x][:4])for x in df.index]

df = pd.merge(left=df, right=df_seeds, how='left', on='TeamName')
df = pd.merge(left=df, right=df_agg, how='left', left_on='TeamName',right_on='T1_TeamName')

df = df[['TeamName','Season','factor1','factor2','Win','Score_diff','Seed','Wins','Upset','WonTourney']]
df.columns = ['TeamName','Season','factor1','factor2','RegWins','RegPoint_diff','Seed','TourneyWins','Upset','WonTourney']

df2020 = df[df['Season']==2020].copy()

df.dropna(inplace=True,subset=['Seed'])

df['TourneyWinsScaled'] = df['TourneyWins']/df['TourneyWins'].max()
df['SeedScaled'] = df['Seed'].astype(int)/df['Seed'].astype(int).max()

df.head()
TeamName Season factor1 factor2 RegWins RegPoint_diff Seed TourneyWins Upset WonTourney TourneyWinsScaled SeedScaled
2 Alabama-1985 1985 -44.957447 -11.601643 0.700000 7.400000 07 2.0 None False 0.333333 0.4375
8 Arizona-1985 1985 -59.622589 20.555855 0.666667 6.851852 10 0.0 None False 0.000000 0.6250
11 Arkansas-1985 1985 -49.902199 2.879017 0.636364 3.363636 09 1.0 None False 0.166667 0.5625
14 Auburn-1985 1985 -43.033291 18.392523 0.620690 3.448276 11 2.0 None False 0.333333 0.6875
21 Boston College-1985 1985 -70.389557 10.855381 0.615385 5.038462 11 2.0 None False 0.333333 0.6875

Important: For the following plots T-SNE representations of trained embeddings will be on the left and mean regular season statistics will be on the right.

#collapse_hide

xrange_tsne = [-80,75]
yrange_tsne = [-80,75]
xrange_raw = [-10,30]
yrange_raw = [.2,1.2]

selector = alt.selection_single(empty='all', fields=['TeamName'])

base = alt.Chart(df).mark_point(filled=True,size=50).encode(
    color=alt.condition(selector,
                        alt.Color('WonTourney:N', scale=alt.Scale(scheme='tableau10')),
                        alt.value('lightgray') ),
    order=alt.Order('WonTourney:N', sort='ascending'),
    tooltip=['TeamName','Seed']
).properties(
    width=250,
    height=250
).add_selection(selector).interactive()

chart1 = [alt.X('factor1:Q',
                scale=alt.Scale(domain=xrange_tsne),
                axis=alt.Axis(title='T-SNE factor 1')),
          alt.Y('factor2:Q',
                scale=alt.Scale(domain=yrange_tsne),
                axis=alt.Axis(title='T-SNE factor 2'))
         ]

chart2 = [alt.X('RegPoint_diff:Q',
                scale=alt.Scale(domain=xrange_raw),
                axis=alt.Axis(title='Average Regular Season Point Difference')),
          alt.Y('RegWins:Q',
                scale=alt.Scale(domain=yrange_raw),
                axis=alt.Axis(format='%', title='Regular Season Win Percentage'))
         ]

base.encode(*chart1)  | base.encode(*chart2)

Tournament winners compared to the field (above): One interesting insight below is how significantly different the 1985 Villanova team is from the other tournament winners. Multiple websites (like this one) list the 1985 Villanova team winning the championship as one of the greatest underdog stories ever. This is far more evident in the T-SNE representation of the embeddings than the plots of win percentage vs. points.

#collapse_hide
xrange_tsne=[-80,100]
yrange_tsne=[-80,100]
xrange_raw=[-20,40]
yrange_raw=[.2,1.2]


selector = alt.selection_single(empty='all', fields=['TeamName'])

base = alt.Chart(df).mark_point(filled=True,size=35).encode(
    color=alt.condition(selector,
                        alt.Color('Seed:Q', scale=alt.Scale(scheme='viridis',reverse=True)),
                        alt.value('lightgray') ),
    order=alt.Order('Seed:Q', sort='descending'),
    tooltip=['TeamName','Seed']
).properties(
    width=250,
    height=250
).add_selection(selector).interactive()

chart1 = [alt.X('factor1:Q',
                scale=alt.Scale(domain=xrange_tsne),
                axis=alt.Axis(title='T-SNE factor 1')),
          alt.Y('factor2:Q',
                scale=alt.Scale(domain=yrange_tsne),
                axis=alt.Axis(title='T-SNE factor 2'))
         ]

chart2 = [alt.X('RegPoint_diff:Q',
                scale=alt.Scale(domain=xrange_raw),
                axis=alt.Axis(title='Average Regular Season Point Difference')),
          alt.Y('RegWins:Q',
                scale=alt.Scale(domain=yrange_raw),
                axis=alt.Axis(format='%', title='Regular Season Win Percentage'))
         ]

base.encode(*chart1)  | base.encode(*chart2)

Colored by seed: We see a high correlation between the assigned seed and our embeddings. Our embeddings appear to be a better representation of the seeding than the aggregated statistics, which makes sense since our method uses pair-wise comparisons and effectively accounts for team strength while aggregated statistics do not.

#collapse_hide
selector = alt.selection_single(empty='all', fields=['TeamName'])

base = alt.Chart(df).mark_point(filled=True,size=35).encode(
    color=alt.condition(selector,
                        alt.Color('TourneyWins:Q', scale=alt.Scale(scheme='viridis',reverse=False)),
                        alt.value('lightgray') ),
    order=alt.Order('TourneyWins:Q', sort='ascending'),
    tooltip=['TeamName','Seed']
).properties(
    width=250,
    height=250
).add_selection(selector).interactive()

chart1 = [alt.X('factor1:Q',
                scale=alt.Scale(domain=xrange_tsne),
                axis=alt.Axis(title='T-SNE factor 1')),
          alt.Y('factor2:Q',
                scale=alt.Scale(domain=yrange_tsne),
                axis=alt.Axis(title='T-SNE factor 2'))
         ]

chart2 = [alt.X('RegPoint_diff:Q',
                scale=alt.Scale(domain=xrange_raw),
                axis=alt.Axis(title='Average Regular Season Point Difference')),
          alt.Y('RegWins:Q',
                scale=alt.Scale(domain=yrange_raw),
                axis=alt.Axis(format='%', title='Regular Season Win Percentage'))
         ]

base.encode(*chart1)  | base.encode(*chart2)

Colored by number of NCAA tournament games won that year: The embeddings appear to be far less correlated to the number of games won by tournament. This is logical since, unlike the seeds, this statistic is not at all represented in the training set.

#collapse_hide
selector = alt.selection_single(empty='all', fields=['TeamName'])

base = alt.Chart(df).mark_point(filled=True,size=50).encode(
    color=alt.condition(selector,
                        alt.Color('Upset:N', scale=alt.Scale(scheme='tableau10')),
                        alt.value('lightgray') ),
    order=alt.Order('Upset:N', sort='ascending'),
    tooltip=['TeamName','Seed']
).properties(
    width=250,
    height=250
).add_selection(selector).interactive()

chart1 = [alt.X('factor1:Q',
                scale=alt.Scale(domain=xrange_tsne),
                axis=alt.Axis(title='T-SNE factor 1')),
          alt.Y('factor2:Q',
                scale=alt.Scale(domain=yrange_tsne),
                axis=alt.Axis(title='T-SNE factor 2'))
         ]

chart2 = [alt.X('RegPoint_diff:Q',
                scale=alt.Scale(domain=xrange_raw),
                axis=alt.Axis(title='Average Regular Season Point Difference')),
          alt.Y('RegWins:Q',
                scale=alt.Scale(domain=yrange_raw),
                axis=alt.Axis(format='%', title='Regular Season Win Percentage'))
         ]

base.encode(*chart1)  | base.encode(*chart2)

Biggest upsets - underdogs in red: Generally, the model agrees with the experts. These were upsets and wouldn't have been predicted by this method. If anything this method would likely have predicted no upset with even greater conviction tahn a model trained on just aggregated points and wins. The only exception to this is the 1986 "upset" of Cleveland State over the Indiana Hoosiers. Both the embeddings model and the aggregated statistics indicate that Cleveland State may have been the better team. Perhaps it was an issue of name recognition that lead this to be called an upset?

#collapse_hide
## create slider and brush
select_year = alt.selection_single(
    name='select', fields=['Season'], init={'Season': 1985},
    bind=alt.binding_range(min=1985, max=2019, step=1))

selector = alt.selection_single(empty='all', fields=['TeamName'])

## create base for charts
base = alt.Chart(df).mark_point(filled=True,size=50).encode(
    color=alt.condition(selector,
                        alt.Color('TourneyWins:Q', scale=alt.Scale(scheme='viridis',reverse=False)),
                        alt.value('lightgray') ),
    order=alt.Order('Seed:Q', sort='descending'),
    tooltip=['TeamName','Seed']
).properties(
    width=250,
    height=250
).add_selection(select_year).transform_filter(select_year).add_selection(selector).interactive()

chart1 = [alt.X('factor1:Q',
                scale=alt.Scale(domain=xrange_tsne),
                axis=alt.Axis(title='T-SNE factor 1')),
          alt.Y('factor2:Q',
                scale=alt.Scale(domain=yrange_tsne),
                axis=alt.Axis(title='T-SNE factor 2'))
         ]

chart2 = [alt.X('RegPoint_diff:Q',
                scale=alt.Scale(domain=xrange_raw),
                axis=alt.Axis(title='Average Regular Season Point Difference')),
          alt.Y('RegWins:Q',
                scale=alt.Scale(domain=yrange_raw),
                axis=alt.Axis(format='%', title='Regular Season Win Percentage'))
         ]

base.encode(*chart1)  | base.encode(*chart2)

Number of games won split out by season - yellow dot is tournament winner: The spread of teams is quite variable year to year. Notably, the tournament that the 1985 Villanova team won as a heavy underdog has less spread in the competition than other years.

#collapse_hide
## 2020 plot
selector = alt.selection_single(empty='all', fields=['TeamName'])

base = alt.Chart(df2020).mark_point(filled=True,size=50).encode(
    color=alt.condition(selector,
                        alt.Color('TeamName:N'),
                        alt.value('lightgray')),
    order=alt.Order('RegWins:Q', sort='ascending'),
    tooltip=['TeamName']
).properties(
    width=250,
    height=250
).add_selection(selector).interactive()

chart1 = [alt.X('factor1:Q',
                scale=alt.Scale(domain=[-80,80]),
                axis=alt.Axis(title='T-SNE factor 1')),
          alt.Y('factor2:Q',
                scale=alt.Scale(domain=[-100,70]),
                axis=alt.Axis(title='T-SNE factor 2'))
         ]

chart2 = [alt.X('RegPoint_diff:Q',
                scale=alt.Scale(domain=[-30,30]),
                axis=alt.Axis(title='Average Regular Season Point Difference')),
          alt.Y('RegWins:Q',
                scale=alt.Scale(domain=[0,1]),
                axis=alt.Axis(format='%', title='Regular Season Win Percentage'))
         ]

base.encode(*chart1)  | base.encode(*chart2)

The 2020 field: Just for fun here is a taste of what we missed in 2020!

Conclusions

The embeddings apppear to have learned which teams are better and which are worse. It seems that they are a better representation of true team skill than simple aggregating the statistics used in model training (wins and point differentials). When the time comes to build a model for the 2021 March Madness Kaggle competition, I will likely return to embeddings as an advanced input feature for my final model, which will be trained on real tournament games! Then will be the time to experiment with training the team embeddings on advanced statistics included in the detailed Kaggle data set in place of or in addition to the target features used here.